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Abstract

We present an experimental and theoretical study of long distance optical binding effects acting upon micro-particles placed in a standing wave optical field. In particular we present for the first time quantitatively the binding forces between individual particles for varying inter-particle separations, polarizations and incident angles of the binding beam. Our quantitative experimental data and numerical simulations show that these effects are essentially enhanced due to the presence of a reflective surface in a sample chamber. They also reveal conditions to form stable optically bound clusters of two and three particles in this geometry. We also show that the inter-particle separation in the formed clusters can be controlled by altering the angle of the beam incident upon the sample plane. This demonstrates new perspectives for the generation and control of optically bound soft matter and may be useful to understand various inter-particle effects in the presence of reflective surfaces.

Figures (9)

Fig. 1 A suspension of polystyrene particles of diameter 820 nm was placed between the microscope slide (top) and the cover-glass (bottom). The cover-glass was coated with a system of dielectric layers (SiO2 and TiO2) reflecting 99% of the incoming beam. This incoming wide Gaussian beam responsible for optical binding (green color, vacuum wavelength 532 nm, beam waist radius 20μm placed at the mirror, incident power 600 mW, Coherent Verdi V5). The interference of the incoming and reflected beam creates a standing wave along y axis over the distance of 20μm between the sample boundaries (set by polystyrene spheres of diameter 20μm). The incident angle θ of the binding beam was controlled in the range of 0–4 degrees by the movable mirror. The second laser pathway at 1064 nm (red color, power in the sample 50–150 mW, IPG ILM-10-1064-LP) acts as a time-shared multiple optical trapping system used for precise force measurement of optical binding forces. The optical trap is formed by oil immersion objective (Nikon Plan Apo, oil, NA 1.4, 100x) and positioned by an acousto-optic deflector (AOD IntraAction DTD-276HD6M driven by two synchronized NI PCI - 5412 RF signal generators). A Basler GigE CCD camera Basler piA640*210gm is used to record the position of the particles. The following focal lengths of the lenses are used in the paths: f1 = 100 mm, f2–4 = 300 mm, condenser lens 25 mm, tube lens in the microscope pathway 200 mm.

Fig. 2 Detailed interaction between the dipoles via the scattered light only and the geometry of the configuration with two particles considered. Left: All single-pathway optical interactions between two dipoles A and B – In addition to the single direct interaction between dipoles (A⇌B) three other pathways are considered due to the reflection of the scattered light at the mirror. Middle: Consideration of the mirror images of the dipoles to include the influence of the mirror for the scattered light. Right: Geometry of the configuration with two particles – Green lines denote direction of the incident and the reflected binding beams placed in the y – z plane with the angle of incidence θ. The inter-particle axis is aligned along the z axis and the centre of mass C of the bound particles is placed at (x,y,z) = (0, yC, zC). yC and zC are the coordinates of the centre of mass C measured from the point where the the centre of the incident beam reflects at the mirror surface.

Fig. 3 Dependence of the optical binding force Fbind on the inter-particle separation zsep for s (left) and p (right) polarization of the incident beam. The dashed and the solid curves denote the results with the interaction through the mirror omitted or included, respectively (see Fig. 2). Parameters of the calculations are: polystyrene particles (with 820 nm in diameter and refractive index of 1.59) placed 20 μm above the mirror (yC = 20μm), angle of incidence θ = 0°, power of the laser beam in the colloidal suspension is 600 mW, vacuum laser wavelength is 532 nm, beam waist of the incident Gaussian beam (w0 = 20μm) is placed at the mirror, refractive index of the water medium is 1.33, and the reflectivity of the mirror is 99%.

Fig. 4 Influence of the distance from the mirror yC on the binding force profile Fbind between two particles. Peaks of slow oscillations move towards larger zsep with increasing yC and their magnitude decreases. Without the presence of the mirror (thick dashed red curve) two particles are predominantly attracted (negative sign of the force). In the case of s polarization the amplitude is again much stronger and several stable separations of two particles can be established. The parameters of the calculation are the same as in Fig. 3 except the incident angle θ = 2°.

Fig. 5 Influence of the lateral placement zC of the bound structure with respect to the incident beam on the binding force profile Fbind between two particles. The thick red line corresponds to a symmetric configuration of the particles (zC = 0) with respect to the incident and reflected beams. The parameters of the calculation are the same as in Fig. 3 except for the incident angle that is set to θ = 0° and 2°, respectively. The vertical placement of the bound structure is fixed at yC = 18μm.

Fig. 6 Stable equilibrium positions of two optically bound particles for different incident angles θ. The circle marks ○ denote the most stable equilibrium positions done by the intersection of the slow modulation with zero-force line and the cross marks × denote all the other equilibrium positions caused by the direct lateral binding between the particles for s-polarized incident beam. All the other parameters of the calculation are the same as in Fig. 3.

Fig. 7 Time record of the deviations of j-th particle from its equilibrium position in the cluster of two optically bound particles if the binding laser is turned off (
〈z〉joff, red) or on (
〈z〉jon, green). Each part contains 1000 positions and the transition of the particles to their new positions took about 250 ms and these data were omitted from further processing.

Fig. 8 Comparison of measured (blue curves) and calculated (red curves) binding forces Fbind between two polystyrene particles for s (left column) and p-polarized (right column) incident beam. In the case of s-polarization (mode ⊥: the electric field is polarized perpendicularly to the connecting line between the particles) the binding forces exert strong oscillations with period equal to λ. These effects are suppressed in the || mode because the radiation between the particles is weaker for the p-polarization. The forces were measured for incident angles θexp in the range 0–4 degrees. For the theoretical prediction the assumed distance from the mirror yC and the incident angle θcdm were chosen within the calculated grid to give the best coincidence with the measured values. The remaining parameters of the calculation are the same as in Fig. 3. Note, that due to a malfunction of our motorized stage during the measuring of binding force – p - polarization, θ = 0° – we show for this case data from different experimental series. In this series the separation between the particles varied between (1.8 – 5.5) μm.

Fig. 9 Comparison of binding forces between the left and the middle particle (1 – 2) and the middle and the right particle (2 – 3). In the case of s-polarization (mode ⊥: the electric field is polarized perpendicularly to the connecting line between the particles) the binding forces exert strong oscillations with period equal to λ. These effects are suppressed in the || mode because the radiation between the particles is weaker for the p-polarization. The accordance between the measured values (blue lines) and calculated ones (red lines) is getting worse for larger angles of incidence θ.